A Collation and Analysis of Two-Dimensional Unsplit Conservative Advection Methods for Volume of Fluid at Interfaces

abstract: The goal of this paper was to do an analysis of two-dimensional unsplit mass and momentum conserving Finite Volume Methods for Advection for Volume of Fluid Fields with interfaces and validating their rates of convergence. Specifically three unsplit transport methods and one split transport method were amalgamated individually with four Piece-wise Linear Reconstruction Schemes (PLIC) i.e. Unsplit Eulerian Advection (UEA) by Owkes and Desjardins (2014), Unsplit Lagrangian Advection (ULA) by Yang et al. (2010), Split Lagrangian Advection (SLA) by Scardovelli and Zaleski (2003) and Unsplit Averaged Eulerian-Lagrangian Advection (UAELA) with two Finite Difference Methods by Parker and Youngs (1992) and two Error Minimization Methods by Pilliod Jr and Puckett (2004). The observed order of accuracy was first order in all cases except when unsplit methods and error minimization methods were used consecutively in each iteration, which resulted in second-order accuracy on the shape error convergence. The Averaged Unsplit Eulerian-Lagrangian Advection (AUELA) did produce first-order accuracy but that was due to a temporal error in the numerical setup. The main unsplit methods, Unsplit Eulerian Advection (UEA) and Unsplit Lagrangian Advection (ULA), preserve mass and momentum and require geometric clipping to solve two-phase fluid flows. The Unsplit Lagrangian Advection (ULA) can allow for small divergence in the velocity field perhaps saving time on the iterative solver of the variable coefficient Poisson System.Dissertation/ThesisMasters Thesis Mechanical Engineering 201

Unstructured un-split geometrical Volume-of-Fluid methods -- A review

Geometrical Volume-of-Fluid (VoF) methods mainly support structured meshes, and only a small number of contributions in the scientific literature report results with unstructured meshes and three spatial dimensions. Unstructured meshes are traditionally used for handling geometrically complex solution domains that are prevalent when simulating problems of industrial relevance. However, three-dimensional geometrical operations are significantly more complex than their two-dimensional counterparts, which is confirmed by the ratio of publications with three-dimensional results on unstructured meshes to publications with two-dimensional results or support for structured meshes. Additionally, unstructured meshes present challenges in serial and parallel computational efficiency, accuracy, implementation complexity, and robustness. Ongoing research is still very active, focusing on different issues: interface positioning in general polyhedra, estimation of interface normal vectors, advection accuracy, and parallel and serial computational efficiency. This survey tries to give a complete and critical overview of classical, as well as contemporary geometrical VOF methods with concise explanations of the underlying ideas and sub-algorithms, focusing primarily on unstructured meshes and three dimensional calculations. Reviewed methods are listed in historical order and compared in terms of accuracy and computational efficiency

Simulating interfacial flows: a farewell to planes

Over the past decades, the volume-of-fluid (VOF) method has been the method of choice for simulating atomization processes, owing to its unique ability to discretely conserve mass. Current state-of-the-art VOF methods, however, rely on the piecewise-linear interface calculation (PLIC) to represent the interface used when calculating advection fluxes. This renders the estimated curvature of the transported interface zeroth-order accurate at best, adversely impacting the simulation of surface-tension-driven flows. In the past few years, there have been several attempts at using piecewise-parabolic interface approximations instead of piecewise-linear ones for computing advection fluxes, albeit all limited to two-dimensional cases or not inherently mass conservative. In this contribution, we present our most recent work on three-dimensional piecewise-parabolic interface reconstruction and apply it in the context of the VOF method. As a result of increasing the order of the interface representation, the reconstruction of the interface and the estimation of its curvature now become a single step instead of two separate ones. The performance of this new approach is assessed both in terms of accuracy and stability and compared to the classical PLIC-VOF approach on a range of canonical test-cases and cases of surface-tension-driven instabilities

Accurate and efficient surface reconstruction from volume fraction data on general meshes

Simulations involving free surfaces and fluid interfaces are important in many areas of engineering. There is, however, still a need for improved simulation methods. Recently, a new efficient geometric VOF method called isoAdvector for general polyhedral meshes was published. We investigate the interface reconstruction step of isoAdvector, and demonstrate that especially for unstructured meshes the applied isosurface based approach can lead to noisy interface orientations. We then introduce a novel computational interface reconstruction scheme based on calculation of a reconstructed distance function (RDF). By iterating over the RDF calculation and interface reconstruction, we obtain second order convergence of both the interface normal and position within cells even with a strict $L_{\infty}$ error norm. In 2D this is verified with reconstruction of a circle on Cartesian meshes and on unstructured triangular and polygonal prism meshes. In 3D the second order convergence is verified with reconstruction of a sphere on Cartesian meshes and on unstructured tetrahedral and polyhedral meshes. The new scheme is combined with the interface advection step of the isoAdvector algorithm. Significantly reduced absolute advection errors are obtained, and for CFL number 0.2 and below we demonstrate second order convergence on all the mentioned mesh types in 2D and 3D. The implementation of the proposed interface reconstruction schemes is straightforward and the computational cost is significantly reduced compared to contemporary methods. The schemes are implemented as an extension to the Computational Fluid Dynamics (CFD) Open Source software package, OpenFOAM. The extension module and all test cases presented in this paper are released as open source